The pseudo-effective cone of a non-K\"ahlerian surface and applications

We describe the positive cone and the pseudo-effective cone of a non-K\"ahlerian surface. We use these results for two types of applications: - Describe the set $\sigma(X)$ of possible total Ricci scalars associated with Gauduchon metrics of fixed volume 1 on a fixed non-K\"ahhlerian surface, and decide whether the assignment $X\mapsto\sigma(X)$ is a deformation invariant. - Study the stability of the canonical extension $$0\to {\cal K}_X\to {\cal A}\to{\cal O}_X\to 0$$ of a class VII surface $X$ with positive $b_2$. This extension plays an important role in our strategy to prove the GSS conjecture using gauge theoretical methods.